Partial leverage plot
Encyclopedia
In statistics
, high-leverage points are those that are outlier
s with respect to the independent variables. Leverage point
s are those that cause large changes in the parameter estimates when they are deleted. Although a leverage point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage
is typically defined as the diagonal of the hat matrix
Partial leverage is used to measure the contribution of the individual independent variables to the leverage of each observation. That is, if hi is the ith row of the diagonal of the hat matrix, the partial leverage is a measure of how hi changes as a variable is added to the regression model.
The partial leverage is computed as:
where
Note that the partial leverage is the leverage of the ith point in the partial regression plot
for the jth variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, high-leverage points are those that are outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s with respect to the independent variables. Leverage point
Leverage (statistics)
In statistics, leverage is a term used in connection with regression analysis and, in particular, in analyses aimed at identifying those observations that are far away from corresponding average predictor values...
s are those that cause large changes in the parameter estimates when they are deleted. Although a leverage point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage
Leverage (statistics)
In statistics, leverage is a term used in connection with regression analysis and, in particular, in analyses aimed at identifying those observations that are far away from corresponding average predictor values...
is typically defined as the diagonal of the hat matrix
Hat matrix
In statistics, the hat matrix, H, maps the vector of observed values to the vector of fitted values. It describes the influence each observed value has on each fitted value...
Partial leverage is used to measure the contribution of the individual independent variables to the leverage of each observation. That is, if hi is the ith row of the diagonal of the hat matrix, the partial leverage is a measure of how hi changes as a variable is added to the regression model.
The partial leverage is computed as:
where
- j = index of independent variable
- i = index of observation
- Xj·[j] = residualsErrors and residuals in statisticsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
from regressing Xj against the remaining independent variables
Note that the partial leverage is the leverage of the ith point in the partial regression plot
Partial regression plot
In applied statistics, a partial regression plot attempts to show the effect of adding an additional variable to the model...
for the jth variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures.
See also
- Partial residual plotPartial residual plotIn applied statistics, a partial residual plot is a graphical technique that attempts to show the relationship between a given independent variable and the response variable given that other independent variables are also in the model.-Background:...
- Partial regression plotPartial regression plotIn applied statistics, a partial regression plot attempts to show the effect of adding an additional variable to the model...
- Variance inflation factorVariance inflation factorIn statistics, the variance inflation factor quantifies the severity of multicollinearity in an ordinary least squares regression analysis...
for a multi-linear fit - Scatterplot matrix